Generalization of the Subset Sum Problem and Cubic Forms

A new algorithm is proposed for deciding whether a system of linear equations has a binary solution over a field of zero characteristic. The algorithm is efficient under a certain constraint on the system of equations. This is a special case of an integer programming problem. In the extended version of the subset sum problem, the weight can be positive or negative. The problem under consideration is equivalent to the analysis of solution existence for several instances of this problem simultaneously. New sufficient conditions are found under which the computational complexity of almost all instances of this problem is polynomial. In fact, the algorithm checks the existence of a cubic hypersurface that passes through each vertex of the unit cube, but does not intersect a given affine subspace. Several heuristic algorithms for solving this problem have been known previously. However, the new methods expand the solution possibilities. Although only the solution existence problem is considered in detail, binary search allows one to find a solution, if any.